Can we detect the emotions (or feelings) of a human through conversations with an AI?
I think you are definitely on a very sensible track. No one defines right or wrong in emotion field. It's not hard science. It's all theories.  

I have recently read a paper regarding emotions in Reinforcement Learning (RL). It has explained briefly emotion from 3 perspectives: psychology, neuroscience and computer science. In particular, your way of emotion definition matches one of the categorical emotion theory in the psychology perspective. The other theories in psychology perspective and componential emotion. You can try to implement them and try out which one works well. The paper has also introduced ways to measure the level of emotions (emotion elicitation).

Here is the link for the paper I have mentioned. I am sure you will receive lots of inspiration. I have also written a summary of this paper. Take a look if the original paper is too long to read.

I don't have any concrete solution for implementing. But the general idea is always trying to categorize abstract concepts and quantify them. And try some thing, and iteratively modify and improve it. All the best!